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Physical Review Letters

Green's Functions in the Theory of Many-Fermion Systems

Mark Bolsterli

Published Jan 15, 1960• 5 citations
Publisher

Abstract

It is shown that the usual single-particle Green's function does not give direct information about the lowest excitation energies if there is a pairing energy (of the type familiar in nuclear physics). An alternate Green's function is suggested which will provide such information. (C.J.G.) It is noted that owing to the similarity existing between electromagnetic interaction and the vector part of the weak interaction proposed by Feynman and Gell-Mann, the contribution of the vector part of the weak interaction to mu /sup -/ capture by a proton is closely related to the electromagnetic form factors of the nucleon. Formula for the capture probability calculated with the renormaalized V-A, weak interaction is given. The correction to the contribution of the vector part of the weak interaction due to the charge and magnetic moment distribution of the nucleon is estimated on the basis of the data from electron-nucleon scattering experiments. (auth)

DOI
10.1103/physrevlett.4.82

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It is shown that the usual single-particle Green's function does not give direct information about the lowest excitation energies if there is a pairing energy (of the type familiar in nuclear physics). An alternate Green's function is suggested which will provide such information. (C.J.G.) It is noted that owing to the similarity existing between electromagnetic interaction and the vector part of the weak interaction proposed by Feynman and Gell-Mann, the contribution of the vector part of the weak interaction to mu /sup -/ capture by a proton is closely related to the electromagnetic form factors of the nucleon. Formula for the capture probability calculated with the renormaalized V-A, weak interaction is given. The correction to the contribution of the vector part of the weak interaction due to the charge and magnetic moment distribution of the nucleon is estimated on the basis of the data from electron-nucleon scattering experiments. (auth)
Using the Green's-function technique of Martin, Schwinger, and Puff, a calculation is made of the total linear response of nuclear matter, as measurable, for instance, in an inelastic electron or muon scattering experiment. In the independent-pair approximation used, two contributions stand out: the quasielastic response of properly dressed particles with well-defined lifetimes, and the contribution due to the scattering between those particles. An extensive analysis is made of the importance of the transverse component of the electromagnetic interaction as well as of the two mentioned contributions. The properly defined quasielastic scattering is predicted to dominate the scattering. A comparison is made with previous calculations by Czy\ifmmode \dot{z}\else \.{z}\fi{} and Gottfried.
Recent developments in the many-body theory of interacting fermions are discussed employing a self-consistent Green function approach. This scheme is outlined and its application to nuclear systems is presented. Special attention is paid to the consistent inclusion of short-range and long-range correlations induced by realistic nucleon-nucleon interactions. Such correlations lead to occupation probabilities which deviate from the simple mean-field or shell-model description. The scheme is extended to incorporate relativistic effects. Also applications of field theoretical models for hadrons in a nuclear medium and their relation to QCD are discussed.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
The short-range and tensor components of the bare nucleon-nucleon interaction induce a sizable depletion of low momenta in the ground state of a nuclear many-body system. The self-consistent Green's function method within the ladder approximation provides an ab initio description of correlated nuclear systems that accounts properly for these effects. The momentum distribution predicted by this approach is analyzed in detail, with emphasis on the depletion of the lowest momentum state. The temperature, density, and nucleon asymmetry (isospin) dependence of the depletion of the Fermi sea is clarified. A connection is established between the momentum distribution and the time-ordered components of the self-energy, which allows for an improved interpretation of the results. The dependence on the underlying nucleon-nucleon interaction provides quantitative estimates of the importance of short-range and tensor correlations in nuclear systems.
An equation set describtng single particle excitations for excitation energies which are small compared with the chemical potential of the system can be obtained by investigating the analytical properties of the Green's functions and taking into account pair correlation. The equations are valid for an arbitrary interaction law. Equations are derived which describe the excited states of a system with a finite number of particles with an accuracy to terms of the order of N/sup -//sup 1//sup ///sup 3/ It is indicated how the results can be applied to real nuclei. (auth)
The equation of state of symmetric nuclear matter is addressed starting both from a realistic interaction derived from nucleon-nucleon scattering processes and from a low-momentum effective potential. The approach is based on finite temperature Green’s functions. The internal energy per particle is estimated from the summation of diagrams and through the Galitskii-Koltun’s sum rule. At present, there is not a unique and reliable theory of nuclear interactions. From the study of nucleon-nucleon scattering processes different models for the basic interaction between nucleons have been developed, but difficulties arise when they are used in calculations which aim to reproduce the properties of bound nuclei. These models all reproduce the experimental N-N phase shifts up to the threshold for pion production with high accuracy, but differ in the short-wavelength region, where less experimental constraints are available. In deriving the properties of dense nuclear matter, if we start from a bare N-N potential we have to take into account the complex correlations induced